Conditional/IF-THEN

A conditional tells us that if the first part is true, the second part must be true as well.

We use => as the conditional operator. So P => Q is read “if P, then Q.”

If-then statements are the most common version of conditional statements. However, “only if,” “required,” “necessary,” and “sufficient” are indicators of alternative ways of making if-then statements. To translate these into the correct conditional statement, think about how to rewrite them in English as if-then statements while keeping the intended meaning. From there, it is easy to translate them into the correct logical conditional.

In P => Q, the P part is sometimes called the antecedent or sufficient condition; the Q part is sometimes called the consequent or necessary condition.

If the antecedent is false, we define the conditional as being vacuously true. The reason for this convention will become clear when we begin discussing truth tables.

P => Q is logically equivalent to ~P v Q. (We will see why when we cover replacement rules.) This means that the conditional is technically unnecessary notation.